Journal de Théorie des Nombres de Bordeaux

Eren Mehmet Kıral

Ian Petrow

Matthew P. Young

Oscillatory integrals with uniformity in parameters

Ralph Kritzinger

Friedrich Pillichshammer

Digital nets in dimension two with the optimal order of $L_p$ discrepancy

Isabel Vogt

Isabel VOGT
Abelian varieties isogenous to a power of an elliptic curve over a Galois
extension
Tome 31, no 1 (2019), p. 205-213.
<http://jtnb.cedram.org/item?id=JTNB_2019__31_1_205_0>
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Journal de Théorie des Nombres
de Bordeaux 31 (2019), 205–213
Abelian varieties isogenous to a power of an
elliptic curve over a Galois extension
par Isabel VOGT
Résumé. Soient E/k une courbe elliptique et k′/k une extension de Galois.
On construit un foncteur exact de la catégorie des modules sans torsion sur
l’anneau des endomorphismes EndEk′ munis d’une action semi-linéaire de
Gal(k′/k) vers la catégorie des variétés algébriques sur k qui sont k′-isogènes
à une puissance de E. Comme application, on donne une preuve simple du
fait que toute courbe elliptique sur k qui est géométriquement à multiplication
complexe, est isogène sur k à une courbe elliptique à multiplication complexe
par un ordre maximal.
Abstract. Given an elliptic curve E/k and a Galois extension k′/k, we
construct an exact functor from torsion-free modules over the endomorphism
ring EndEk′ with a semilinear Gal(k′/k) action to abelian varieties over k that
are k′-isogenous to a power of E. As an application, we give a simple proof that
every elliptic curve with complex multiplication geometrically is isogenous
over the ground field to one with complex multiplication by a maximal order.
1. Introduction
Let E be an elliptic curve over a field k. As in [4], the theory of abelian
varieties isogenous over k to a power of E is related to the theory of finitely
presented torsion-free modules over the endomorphism ring Rk := Endk E.
To recall briefly, there is a functor
HomRk( · , E) :
{
finitely presented
left Rk-modules
}opp
→
{
commutative proper
k-group schemes
}
,
such that for M a finitely presented Rk-module and C a k-scheme, we have
Homk(C,HomRk(M,E)) = HomRk(M,Homk(C,E)).
Restricting to torsion-free modules, we obtain a functor
HomRk( · , E) :
{
fin. pres. tors. free
left Rk-modules
}opp
→
{
abelian k-varieties
isogenous to Er, r ∈ Z
}
.
Manuscrit reçu le 29 avril 2018, accepté le 19 octobre 2018.
2010 Mathematics Subject Classification. 14K02, 11G10.
Mots-clefs. abelian varieties, complex multiplication, isogenies.
This research was partially supported by the National Science Foundation Graduate Research
Fellowship Program under Grant DMS-1122374 as well as Grant DMS-1601946.
206 Isabel Vogt
This functor is fully faithful, but not in general an equivalence of categories.
For example, if chat k = 0 and E does not have complex multiplication,
then the image of this functor consists entirely of the powers of E; yet it is
possible for E to be k-isogenous to a non-isomorphic curve.
In fact, this functor is never surjective if E/k acquires complex mul-
tiplication (CM) over a separable quadratic extension k′/k: E is always
k-isogenous to its k′/k quadratic twist E′/k, as we now show. Suppose that
Ek′ has CM by the order O. Let α ∈ O be a purely imaginary element of
O, so that the nontrivial element σ ∈ Gal(k′/k) acts as σα = −α. Multi-
plication by α composed with the isomorphism Ek′
'−→ E′k′ is Galois stable,
and so descends to an isogeny E → E′ over k.
In this note we describe a generalization of the functor HomRk( · , E),
which in particular addresses the case when Endk E 6= Endk̄ E. We will
show that the essential image of this functor always contains all of the
quadratic twists of E.
More generally, one may also consider abelian varieties, as in the case
of nontrivial twists, that are isogenous to a power of an elliptic curve only
after passing to a Galois extension of the ground field. When restricted to
torsion-free modules, the image of the functor we construct will lie in the
category of abelian varieties that become isogenous to a power of E over
a Galois extension. This functor may therefore shed light upon abelian
varieties that are not isogenous to a power of E over the ground field, and
therefore missed by the previous functor, but become isogenous to a power
of E after making a suitable extension.
Let k′/k be a finite Galois extension and let G := Gal(k′/k). As every
proper commutative group scheme over a field is projective, the category
of commutative proper k-group schemes is equivalent to the category of
commutative proper k′-group schemes equipped with descent data for k′/k.
For this reason we may identify commutative proper k-group schemes with
commutative proper k′-group schemes with an action of G, such that all
maps, including the structure maps, are G-equivariant.
Let R = Rk′ := EndEk′ be the endomorphism ring of the base change
Ek′ of E to k′. In particular, R is noetherian. This inherits an action of G.
We denote this action by r 7→ σr for σ ∈ G and r ∈ R. We may then form
the twisted group ring R〈G〉 as the free R-module
R〈G〉 =
⊕
σ∈G
R · σ
with the commutation relation σr = σrσ. In this way, a module over R〈G〉
is an R-module with a semilinear G-action.
We give a construction of the following in Section 2.
Abelian varieties isogenous to a power of an elliptic curve 207
Theorem 1.1. There exists an exact functor
HomR〈G〉( · , Ek′) :
{
fin. pres. left
R〈G〉-modules
}opp
→
{
commutative proper
k-group schemes
}
,
such that
(1) For a finitely presented R〈G〉-module M and any k-scheme C, we
have
Homk(C,HomR〈G〉(M,Ek′)) = HomR〈G〉(M,Homk′(Ck′ , Ek′)).
(2) The functor agrees with HomR( · , Ek′) under the forgetful functors
mapping R〈G〉-modules to R-modules and base-changing k-schemes
to k′-schemes.
In particular, this shows that if M is an R〈G〉-module that is torsion-
free as an R-module, then HomR〈G〉(M,Ek′) is an abelian variety A over k,
such that Ak′ is isogenous to a power of E. However, we may say something
about the k-isogeny class of A as well. Recall that the category of abelian
varieties over k up to isogeny has the same objects as the category of abelian
varieties over k, but all isogenies are inverted.
Let F = R⊗ZQ. Since F is semisimple, the category of finitely presented
modules over the twisted group algebra F 〈G〉 is semisimple (as in the case
when the action of G on F is trivial, see Lemma 2.3 below). Let S1, . . . , S`
denote the simple objects.
Theorem 1.2. There is a functor
HomF 〈G〉( · , Ek′) :
{
fin. pres.
F 〈G〉-modules
}opp
→
{
abelian varieties over k
up to isogeny
}
,
compatible with the functor HomR〈G〉( · , Ek′). For any torsion-free finitely
presented R〈G〉-module M , HomR〈G〉(M,Ek′) is isogenous over k to a
product of powers of the abelian varieties HomF 〈G〉(Si, Ek′).
As an application of this Galois-equivariant functor, we give a simple and
new proof of the following old result (see [3, Prop. 25], [2, Prop. 2.2], [6,
Prop. 5.3], [5, Prop. 2.3b] for other proofs of the same or similar results),
which is useful in reducing questions about arbitrary CM elliptic curves to
those with complex multiplication by a maximal order; this will be used
in [7] for this very reason.
Corollary 1.3. Let k be a number field and let E/k be an elliptic curve.
Suppose that Ek̄ has complex multiplication by an order O in an imaginary
quadratic field F . Then there exists an elliptic curve E′/k and an isogeny
ϕ : E′ → E
defined over k, such that E′
k̄
has complex multiplication by the full ring of
integers of F .
208 Isabel Vogt
Acknowledgments. I would like to thank Bjorn Poonen for suggesting
that the initial functor could be useful in addressing the problem in Corol-
lary 1.3 and for helpful discussions. I also thank Pavel Etingof for answering
some questions, and Pete Clark for explaining the history of the result in
Corollary 1.3 and providing references.
2. Categorical Constructions
For any R〈G〉-module M , let HomR〈G〉(M,Ek′) be the functor from the
category of k-schemes to the category of sets sending a k-scheme C to
HomR〈G〉(M,Homk′(Ck′ , Ek′)).
Let
Resk′/k : {comm. proper k′-group schemes}→{comm. proper k-group sch}
denote Weil restriction of scalars.
Proposition 2.1. The functor HomR〈G〉(M,Ek′) is representable by a
commutative proper k-group scheme. When M = R〈G〉, the representing
scheme is Resk′/k Ek′.
Proof. We first show that HomR〈G〉(R〈G〉, Ek′) = Resk′/k Ek′ . Indeed, by
the universal property of the restriction of scalars we have
Homk(C,Resk′/k Ek′)
= Homk′(Ck′ , Ek′) = HomR〈G〉(R〈G〉,Homk′(Ck′ , Ek′))
= Homk(C,HomR〈G〉(R〈G〉, Ek′))
as desired. Note that the k-scheme Resk′/k Ek′ has endomorphisms by R〈G〉.
Now let
(2.1) R〈G〉m → R〈G〉n →M → 0
be a finite presentation of M as R〈G〉-modules. The map R〈G〉m → R〈G〉n
is given by multiplication on the right by a m×n matrix X. Multiplication
by X on the left also defines a map
(Resk′/k Ek′)n → (Resk′/k Ek′)m.
As commutative proper group schemes over k form an abelian category, the
above morphism has a kernel in this category,
(2.2) 0→ A→ (Resk′/k Ek′)n → (Resk′/k Ek′)m.
We claim that A represents the functor HomR〈G〉(M,Ek′) defined above.
Indeed, we may apply the left-exact functor Homk(C,−) to (2.2) to obtain
(2.3) 0→ Homk(C,A)→ Homk′(Ck′ , Ek′)n → Homk′(Ck′ , Ek′)m.
Abelian varieties isogenous to a power of an elliptic curve 209
Similarly apply the left-exact functor HomR〈G〉( · ,Homk′(Ck′ , Ek′)) to (2.1)
to obtain
(2.4) 0→ HomR〈G〉(M,Homk′(Ck′ , Ek′))→ Homk′(Ck′ , Ek′)n
→ Homk′(Ck′ , Ek′)m.
As the right maps in both (2.3) and (2.4) are induced by multiplication by
X, the kernels are functorially isomorphic, as desired. 
Recall that we let F := R ⊗ Q. We now show that we have a compati-
ble functor taking values in the category of abelian varieties over k up to
isogeny, as in Theorem 1.2. For objects B1 and B2 in the isogeny category,
denote by Homisog(B1, B2) the morphisms in the isogeny category.
For a finitely presented F 〈G〉-module N , define the functor
HomF 〈G〉(N,Ek′)
from the isogeny category of abelian varieties over k to sets by
B 7→ HomF 〈G〉(N,Homisog(Bk′ , Ek′)).
Proposition 2.2. The functor HomF 〈G〉(N,Ek′) defined above is repre-
sented by an abelian variety in the isogeny category over k.
Proof. As above, let B be an object of the isogeny category of abelian
varieties over k. If N = M ⊗R〈G〉 F 〈G〉, we have
HomF 〈G〉(N,Homisog(Bk′ , Ek′))
= HomF 〈G〉(M ⊗ F 〈G〉,Homk′(Bk′ , Ek′)⊗Q),
= HomR〈G〉(M,R〈G〉Homk′(Bk′ , Ek′)⊗Q),
= HomR〈G〉(M,Homk′(Bk′ , Ek′))⊗Q.
And so this functor agrees with the original HomR〈G〉( · , Ek′) after com-
posing with the localization map to the isogeny category.
Therefore the functor HomF 〈G〉(F 〈G〉, Ek′) is represented by Resk′/k(Ek′)
in the isogeny category. Furthermore, as the isogeny category is an abelian
category, the same proof as in Lemma 2.1 shows that for all finitely pre-
sented F 〈G〉-modules N , the functor HomF 〈G〉(N,Ek′) is represented by
an object in the isogeny category. 
Lemma 2.3. Let F be a semisimple Q-algebra. Then F 〈G〉 is semisimple.
Proof. Let V →W be a surjection of finite-dimensional left F 〈G〉-modules.
As F is semisimple, we may choose a splitting ϕ : W → V as F -modules.
The map π : W → V given by
π(w) =
∑
g∈G
gϕ(g−1w),
defines a splitting as F 〈G〉-modules. 
210 Isabel Vogt
Since the endomorphism algebra of an elliptic curve is always semisimple,
Lemma 2.3 combined with the above construction proves Theorem 1.2.
We now show compatibility with base change and restriction of scalars.
Lemma 2.4. Let M be a finitely presented left R-module and set A :=
HomR(M,Ek′). Then we have that
Resk′/k A 'HomR〈G〉(R〈G〉 ⊗RM,Ek′).
Proof. By Yoneda’s Lemma, it suffices to show that Resk′/k A and
HomR〈G〉(R〈G〉 ⊗RM,Ek′
have the same functors of points. Let C be a k-scheme. By the universal
property of restriction of scalars, we have
Resk′/k A(C) = A(Ck′) = HomR(M,Hom(Ck′ , Ek′)).
By the adjunction between restriction and induction [1, Prop. 2.8.3(i)], we
have
HomR(M,Hom(Ck′ , Ek′)) ' HomR〈G〉(R〈G〉 ⊗M,Hom(Ck′ , Ek′)),
which completes the proof. 
For any ring R and any left R-algebra S, let S denote the (R,S)-bimodule
S under multiplication by R on the left and S on the right.
Lemma 2.5. Let M be a finitely presented left R〈G〉-module and let A :=
HomR〈G〉(M,Ek′). Then the base change Ak′ is isomorphic to
HomR(RM,Ek′),
where RM denotes the underlying R-module of M .
Proof. By Yoneda’s Lemma, it suffices to show that Ak′ and HomR(RM,
Ek′) have the same functor of points. Let D be a k′-scheme. Let kD denote
the k-scheme whose structure morphism is the composition D → Spec k′ →
Spec k. By the universal property of fiber products,
Ak′(D) = A(kD) = HomR〈G〉
(
M,Ek′
(
(kD)k′
))
.
We are thus reduced to showing that
HomR〈G〉
(
M,Ek′
(
(kD)k′
))
= HomR(RM,Ek′(D)).
Furthermore, by the adjunction between restriction and coinduction [1,
Prop. 2.8.3(ii)],
HomR(RM,Ek′(D)) = HomR〈G〉(M,HomR(R〈G〉, Ek′(D))).
It suffices then to show that, as left R〈G〉-modules,
Ek′
(
(kD)k′
)
' HomR(R〈G〉, Ek′(D)).
Abelian varieties isogenous to a power of an elliptic curve 211
As k′ ⊗k k′ '
∏
σ∈G k
′, we have that
(kD)k′ ' qσ∈GD,
where τ ∈ G maps the σth copy of D to the τσth copy. Elements of
Ek′ (qσ∈GD) can be represented as tuples f = (fσ)σ where fσ ∈ Ek′(D)
and
(τf)σ = τfτ−1σ.
Similarly elements of HomR(R〈G〉, Ek′(D)) are tuples g = (gσ)σ where
g ∈ Ek′(D) and (τg)σ = gστ . The correspondence
gσ = σfσ−1 ,
shows that these R〈G〉-modules are isomorphic. 
Note that any finitely presented R-module with a semilinear G-action is
also a finitely presented R〈G〉-module, as we now explain. As R is noether-
ian, it suffices to show that any R〈G〉-moduleM , which is finitely generated
as an R-module, admits a G-equivariant surjection from R〈G〉n for some n.
If m1, . . . ,mr are R-module generators of M , then the map R〈G〉r → M
sending ei 7→ mi has the desired properties.
We have the following nice properties of the functor HomR〈G〉( · , Ek′).
Proposition 2.6. Let E/k be an elliptic curve and k′/k a finite Galois
extension with Galois group G. Let R := Endk′ E and let M be a finitely
presented R〈G〉-module and let A := HomR〈G〉(M,Ek′).
(1) HomR〈G〉( · , Ek′) is exact.
(2) A is a commutative proper group scheme over k of dimension rkR(M).
(3) If M is torsion-free as an R-module, then A is an abelian variety
over k such that Ak′ is isogenous to a power of Ek′.
(4) HomR〈G〉(R,Ek′) = E.
Proof. Parts (1)–(3) follow from the corresponding properties after base-
extension to k′ [4, ThM. 4.4]. For part (4), we have
Homk(C,HomR〈G〉(R,Ek′)) = HomR〈G〉(R,Homk′(Ck′ , Ek′))
= Homk′(Ck′ , Ek′)G
= Homk(C,E). 
3. Examples and Applications
As an example of Theorem 1.2, we have the following in the special case
k′/k is a separable quadratic extension.
Proposition 3.1. Let k′/k be a separable quadratic extension and set G :=
Gal(k′/k) with nontrivial element σ. Let E′ denote the corresponding k′/k
quadratic twist of E. If M is a torsion-free R〈G〉-module, then the abelian
variety HomR〈G〉(M,Ek′) is isogenous to Er × (E′)r
′
for some r, r′ ≥ 0.
212 Isabel Vogt
Proof. As above, we denote the G-action on F by σ(x) = σx. By Theo-
rem 1.2, it suffices to decompose F 〈G〉 into simple modules. Let σF denote
F endowed with the G-action σ(x) = −σx for x ∈ F . Then we have an
isomorphism
F ⊕ σF ϕ−→ F 〈G〉,
defined by ϕ(a, b) = a(σ + 1) + b(σ − 1). By comparing functors of points,
we have HomF 〈G〉(F,Ek′) ' E and HomF 〈G〉(σF,Ek′) ' E′ in the isogeny
category. 
Example 3.2. Consider the special case that k is a number field, and
E has complex multiplication by an order O in the imaginary quadratic
field F = FracO, which is defined over the quadratic extension k′ = kF/k
with Galois group G = {1, σ}. Let M be a finitely presented O〈G〉-module
that is torsion-free as an O-module. Then as multiplication by a totally
imaginary element of O defines an isomorphism σF ' F of F 〈G〉-modules,
we have that HomO〈G〉(M,Ek′) is an abelian variety defined over k, which
is isogenous over k to a power of E.
Continuing in the setup of the previous example, we conclude by proving
Corollary 1.3.
Proof of Corollary 1.3. This follows from Example 3.2 as the full ring of
integers OF of F is a finitely presented O〈G〉-module, which is torsion-
free of rank 1 over O. The full endomorphism ring of E is defined over
(the at most quadratic extension) k′, so we have that O = EndEk′ . This
inherits a (possibly trivial) action of G. We therefore have the following
exact sequence of O〈G〉-modules:
0→ O → OF → OF /O → 0,
where the action of G is induced from the action on F . Applying the functor
HomO〈G〉( · , Ek′) we obtain an exact sequence
0→HomO〈G〉(OF /O, Ek′)→ E′ → E → 0,
where E′ is again an elliptic curve over k and the right map is an isogeny
to E. By functoriality, E′k′ has an action of OF , as desired. 
References
[1] D. J. Benson, Representations and cohomology. I. Basic representation theory of finite
groups and associative algebras, Cambridge Studies in Advanced Mathematics, vol. 30, Cam-
bridge University Press, 1998.
[2] A. Bourdon & P. Pollack, “Torsion subgroups of CM elliptic curves over odd degree
number fields”, Int. Math. Res. Not. 2017 (2017), no. 16, p. 4923-4961.
[3] P. L. Clark, B. Cook & J. Stankewicz, “Torsion points on elliptic curves with complex
multiplication”, Int. J. Number Theory 9 (2013), no. 2, p. 447-479.
Abelian varieties isogenous to a power of an elliptic curve 213
[4] B. W. Jordan, A. G. Keeton, B. Poonen, E. M. Rains, N. Shepherd-Barron & J. T.
Tate, “Abelian varieties isogenous to a power of an elliptic curve”, Compos. Math. 154
(2018), no. 5, p. 934-959.
[5] S. Kwon, “Degree of isogenies of elliptic curves with complex multiplication”, J. Korean
Math. Soc. 36 (1999), no. 5, p. 945-958.
[6] K. Rubin, “Elliptic curves with complex multiplication and the conjecture of Birch and
Swinnerton–Dyer”, in Arithmetic theory of elliptic curves (Cetraro, 1997), Lecture Notes in
Mathematics, vol. 1716, Springer, 1997, p. 167-234.
[7] I. Vogt, “A local-global principle for isogenies of composite degree”, https://arxiv.org/
abs/1801.05355, 2018.
Isabel Vogt
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139, USA
E-mail: ivogt@mit.edu
URL: http://web.mit.edu/~ivogt/


Abelian varieties isogenous to a power of an elliptic curve over a Galois extension

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Higher congruences between newforms and Eisenstein series of squarefree level

Peter Humphries

Snehal M. Shekatkar

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Biases in prime factorizations and Liouville functions for arithmetic progressions

Oleg Karpenkov

Matty van-Son

Generalized Perron Identity for broken lines

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Michael M. Schein

On the flat cohomology of binary norm forms

Jian Wang

Jian WANG
On the cyclic torsion of elliptic curves over cubic number fields (II)
Tome 31, no 3 (2019), p. 663-670.
<http://jtnb.centre-mersenne.org/item?id=JTNB_2019__31_3_663_0>
© Société Arithmétique de Bordeaux, 2019, tous droits réservés.
L’accès aux articles de la revue « Journal de Théorie des Nom-
bres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique
l’accord avec les conditions générales d’utilisation (http://jtnb.
centre-mersenne.org/legal/). Toute reproduction en tout ou partie
de cet article sous quelque forme que ce soit pour tout usage autre
que l’utilisation à fin strictement personnelle du copiste est con-
stitutive d’une infraction pénale. Toute copie ou impression de ce
fichier doit contenir la présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.centre-mersenne.org/
Journal de Théorie des Nombres
de Bordeaux 31 (2019), 663–670
On the cyclic torsion of elliptic curves over cubic
number fields (II)
par Jian WANG
Résumé. Le résultat de Merel sur la forme forte de la conjecture de borne
uniforme a mis en valeur la classification des parties de torsion des groupes de
Mordell–Weil des courbes elliptiques définies sur les corps de nombres de degré
fixé d. Dans cet article, nous étudions les sous-groupes de torsion cycliques
des courbes elliptiques sur les corps de nombres cubiques. Pour N = 49, 40, 25
ou 22, nous montrons que Z/NZ n’est pas un sous-groupe de E(K)tor pour
toute courbe elliptique E sur un corps de nombres cubique K.
Abstract. Merel’s result on the strong uniform boundedness conjecture
made it meaningful to classify the torsion part of the Mordell–Weil groups of
all elliptic curves defined over number fields of fixed degree d. In this paper,
we discuss the cyclic torsion subgroup of elliptic curves over cubic number
fields. For N = 49, 40, 25 or 22, we show that Z/NZ is not a subgroup of
E(K)tor for any elliptic curve E over a cubic number field K.
1. Introduction
In 1996, Merel [16] finally proved the strong uniform boundedness con-
jecture for elliptic curves over number fields.
Theorem 1.1 (Merel). For every positive integer d, there exists an integer
Bd such that for every number field K of degree d and every elliptic curve
E over K, we always have
|E(K)tor| ≤ Bd
Merel’s result made it meaningful to classify the torsion part of the
Mordell–Weil groups of all elliptic curves defined over number fields of
fixed degree d. The case d = 1 was solved by Mazur [15] and Kubert [13].
The case d = 2 was solved by Kamienny [9], Kenku and Momose [12].
In 2016, Derickx and the group of Etropolski, Morrow and Zureick-Brown
each announced a solution of the case d = 3 [4], but there is still no publicly
available preprint until April 2019. In view of the work of Parent [20, 21],
Jeon–Kim–Schweizer [8] and Najman [17], we conjectured (see [27]) that
Manuscrit reçu le 16 décembre 2018, révisé le 13 avril 2019, accepté le 11 mai 2019.
2010 Mathematics Subject Classification. 11G05, 11G18.
Mots-clefs. torsion subgroup, elliptic curves, modular curves.
664 Jian Wang
Z/NZ is not a cyclic torsion subgroup of the Mordell–Weil group of any
elliptic curve over a cubic number field for 24 values of N :
22, 24, 25, 26, 27, 28, 30, 32, 33, 35, 36, 39, 40,
42, 45, 49, 55, 63, 65, 77, 91, 121, 143, 169.
In [27], six (N = 55, 65, 77, 91, 143, 169) of the above values were ruled
out using refinements of a criterion originally due to Kamienny. In this
paper, we rule out four further values. Namely we obtain the following:
Theorem 1.2. If N = 49, 40, 25 or 22, then Z/NZ is not a subgroup of
E(K)tor for any elliptic curve E over a cubic number field K.
Acknowledgments. We thank Sheldon Kamienny for valuable ideas and
insightful comments. We thank the anonymous referee for helpful comments
and suggestions.
2. Preliminaries
In this section, we omit the background materials which were covered in
Section 2 of [27]. Readers who are interested may switch there.
Let N be a positive integer. Let X1(N) (resp. X0(N)) be the modular
curve defined over Q associated to the congruence subgroup Γ1(N) (resp.
Γ0(N)). We denote by Y1(N) = X1(N)\{cusps}, Y0(N) = X0(N)\{cusps}
the corresponding affine curves. Denote by J1(N) (resp. J0(N)) the Jaco-
bian of X1(N) (resp. X0(N)).
For a modular curve X, let X(d) be the d-th symmetric power of X. Sup-
pose K is a number field of degree d over Q and x ∈ X(K). Let x1, . . . , xd
be the images of x under the distinct embeddings τi : K ↪→ C, 1 ≤ i ≤ d.
Define
Φ : X(d) −→ JX
by Φ(P1 + · · ·+Pd) = [P1 + · · ·+Pd− d∞] where JX is the Jacobian of X,
and [ · ] denotes the divisor class. The following lemma of Frey [5] plays an
important role in Lemma 3.5.
Lemma 2.1 (Frey). Let K be a number field. If Φ|X(d)(K) is not injective,
then there is a K-rational covering π : X −→ P1K of degree ≤ d.
For a modular curve X over C, X is called d-gonal if there exists a finite
C-morphism π : X −→ P1C of degree d. The smallest possible d is called
the C-gonality of X, denoted by GonC(X). Since the cusp at infinity is a
rational point, X1(N) is 1-gonal if and only if N is among the eleven values
N = 1−10, 12 with genus 0. The complete lists of 2-gonal and 3-gonal ones
were determined by Ishii–Momose [7] and Jeon–Kim–Schweizer [8].
cyclic torsion of elliptic curves 665
Proposition 2.2 (Ishii–Momose). The modular curve X1(N) is 2-gonal if
and only if N is one of the following:
N = 1− 10, 12 (g = 0);
N = 11, 14, 15 (g = 1);
N = 13, 16, 18 (g = 2).
Proposition 2.3 (Jeon–Kim–Schweizer). The modular curve X1(N) is 3-
gonal if and only if N is one of the following:
N = 1− 10, 12 (g = 0);
N = 11, 14, 15 (g = 1);
N = 13, 16, 18 (g = 2);
N = 20 (g = 3).
Any noncuspidal point of X1(N) is represented by (E,±P ), where E is
an elliptic curve and P ∈ E is a point of order N . Any noncuspidal point
of X0(N) is represented by (E,C), where E is an elliptic curve and C ⊂ E
is a cyclic subgroup of order N . The map π : X1(N) −→ X0(N) sends
(E,±P ) to (E, 〈P 〉), where 〈P 〉 is the cyclic subgroup generated by P .
Let p be a prime such that p - N . Igusa’s theorem [6] says that the
modular curves X1(N) and X0(N) have good reduction at prime p. The
following theorem of Serre and Milne says that reducing the modular curve
is compatible with reducing the modular interpretation.
Theorem 2.4 (Serre–Milne [19, Theorem 1]). Any point of Y1(N) or
Y0(N), rational over a field K (of characteristic not dividing N), is rep-
resented by a K-rational pair (i.e. E is defined over K, and P is rational
over K, or C is a group rational over K), and conversely.
Let K be a number field with ring of integers OK , ℘ ⊂ OK a prime ideal
lying above p, k = Fq = OK/℘ its residue field. Let E be an elliptic curve
over K and P ∈ E(K) a point of order N . Let Ẽ be the fibre over k of the
Néron model of E, and let P̃ ∈ Ẽ(k) be the reduction of P . The following
theorem (see, for instance, [1, §7.3 Proposition 3]) shows that P̃ has order
N when p - N .
Theorem 2.5. Let m be a positive integer relatively prime to char(k).
Then the reduction map
E(K)[m] −→ Ẽ(k)
is injective.
666 Jian Wang
Let k = Fq be the finite field with q = pn elements. Let E/k be an elliptic
curve over k. Let |E(k)| be the number of points of E over k. Then Hasse’s
theorem states that
||E(k)| − q − 1| ≤ 2√q
i.e.
(1−
√
pn)2 ≤ |E(k)| ≤ (1 +
√
pn)2
Let t = q+1−|E(k)|, E is called ordinary if (t, q) = 1, otherwise it is called
supersingular. In the range proposed by Hasse’s theorem, all the ordinary t
appear, while the supersingular t only appears in restricted case. This was
determined by Waterhouse [28, Theorem 4.1]:
Proposition 2.6 (Waterhouse). The isogeny classes of elliptic curves over
k are in one-to-one correspondence with the rational integers t having |t| ≤
2√q and satisfying one of the following conditions:
(1) (t, p) = 1;
(2) If n is even: t = ±2√q;
(3) If n is even and p 6≡ 1 mod 3: t = ±√q;
(4) If n is odd and p = 2 or 3: t = ±p
n+1
2 ;
(5) If either (i) n is odd or (ii) n is even and p 6≡ 1 mod 4: t = 0.
3. Method
The following Theorem states that the Jacobian J1(N) decomposes to a
direct sum of modular abelian varieties.
Theorem 3.1. [2, Theorem 6.6.6] The Jacobian J1(N) is isogenous to a
direct sum of abelian varieties (over Q) associated to equavalence classes of
newforms
J1(N) −→
⊕
f
A
mf
f
with f(τ) =
∑∞
n=1 an(f)e2πinτ newforms of level dividing N .
The L-series L(Af , s) of Af coincides, up to a finite number of Euler
factors, with the product∏
σ
L(fσ, s) =
∏
σ
∞∑
n=1
aσnn
−s
where σ runs through embeddings σ : Kf ↪−→ C with Kf = Q({an}) the
number field of f (See [23, §7.5]). The following proposition is a special
case of Corollary 14.3 in Kato [10]:
Proposition 3.2. Let A be an abelian variety over Q such that there is a
surjective homomorphism J1(N) −→ A for some N ≥ 1. If L(A, 1) 6= 0,
then A(Q) is finite.
cyclic torsion of elliptic curves 667
The decomposition of J1(N) and the non-vanishing of the L-series at
s = 1 of modular abelian varieties can be calculated in Magma [26]. If
L(Af , 1) 6= 0 for all Af , then we know Af (Q) is finite for all Af , therefore
J1(N)/Q is finite. For the N ≤ 65 in the list in [27], Table 3.1 is the result of
calculations in Magma. The second column t is the number of non-isogenous
modular abelian varieties in the decompositon J1(N) =
⊕t
i=1A
mi
i . The
third column list the dimension di and multiplicity mi of each Ai (we omit
mi if mi = 1). The fourth column verifies non-vanishing of L-series at 1 (a
mark T means L(Ai, 1) 6= 0 is verified, otherwise we place a mark F).
The results in Table 3.1 verifies that J1(N)(Q) is finite for the first 17
values (N ≤ 55) in the list of Section 1. For N = 65, 63, we don’t know
whether J1(N)(Q) is finite or not since we fail to verify the non-vanishing
at 1 of the L-series of one of its quotients.
Table 3.1. Decompostion of J1(N)
N t di(mi) L(Ai, 1) 6= 0
49 5 1, 48, 6, 12, 2 T, T, T, T, T
25 2 8, 4 T, T
27 2 1, 12 T, T
32 4 1, 4, 8, 2(2) T, T, T, T
65 19 1, 2, 2, 6, 20, 20, 8, 2(2), 8, F , T, T, T, T, T, T, T, T,
2, 8, 2, 8, 4, 4, 12, 6, 2, 2 T, T, T, T, T, T, T, T, T, T
39 10 1, 2, 4, 8, 2, 2(2), 4, 2, 4, 2 T, T, T, T, T, T, T, T, T, T
26 5 1, 1, 2(2), 2, 2 T, T, T, T, T
55 10 1, 2, 1(2), 4, 32, 8, 8, 16, 4, 4 T, T, T, T, T, T, T, T, T, T
33 6 1, 1(2), 8, 4, 4, 2 T, T, T, T, T, T
22 2 1(2), 4 T, T
35 8 1, 2, 2, 4, 4, 4, 4, 4 T, T, T, T, T, T, T, T
63 20 1, 2, 1(2), 6, 6, 2, 10, 4, 2(2), 2, T, T, T, T, T, T, T, T, T,F
10, 2, 2, 2(2), 2, 10, 2, 10, 12, 4 T, T, T, T, T, T, T, T, T, T
28 4 1(2), 4, 2, 2 T, T, T, T
45 8 1, 1(2), 2, 6, 16, 4, 2, 8 T, T, T, T, T, T, T, T
30 4 1, 1(2), 4, 2 T, T, T, T
40 7 1, 1(2), 4, 2(2), 8, 2, 4 T, T, T, T, T, T, T
36 5 1, 8, 2, 2(2), 2 T, T, T, T, T
24 3 1, 2, 2 T, T, T
The famous specialization lemma is an immediate consequence of the
classification of Oort–Tate [25] on finite flat group schemes of rank p (this
result was generalized to finite flat group schemes of type (p, . . . , p) by
Raynaud [22]). If the group scheme is contained in an abelian variety, as
668 Jian Wang
we stated in the following, the specialization lemma follows from elemen-
tary properties of formal Lie groups (see, for example, the Appendix of
Katz [11]).
Lemma 3.3 (Specialization Lemma). Let K be a number field. Let ℘ ⊂ OK
be a prime above p. Let A/K be an abelian variety. Suppose the ramification
index e℘(K/Q) < p− 1. Then the reduction map
Ψ : A(K)tor −→ A(Fp)
is injective.
In the proof of Lemma 3.5, we also use a theorem of Manin [14] and
Drinfeld [3].
Theorem 3.4 (Manin–Drinfeld). Let Γ ⊂ SL2(Z)/(±1) be a congruence
subgroup. x, y ∈ P1(Q) and x, y the images of x and y respectively, on H/Γ.
Then the class of divisors (x)− (y) on curve H/Γ has finite order.
Lemma 3.5. SupposeN > 4 such that GonC(X1(N)) > d, J1(N)(Q) is
finite, p > 2 is a prime not dividing N . Let K be a number field of degree
d over Q and ℘ a prime of K over p. Let E/K be an elliptic curve with a
K-rational point P of order N , i.e. x = (E,±P ) ∈ Y1(N)(K). Then E has
good reduction at ℘.
Proof. Suppose E has additive reduction at ℘, then Ẽ(k)0 ∼= Ga/k with
|Ga/k| = pi, i ≤ d and Ẽ(k)/Ẽ(k)0 ∼= G with |G| ≤ 4 (this comes from the
Kodaira–Néron classification of special fibers on Néron models of elliptic
curves. see for example [24, §IV.8]). Since P̃ is a k-rational point of order
N in Ẽ, then N divides |Ẽ(k)| = |Ga/k||G|, which is impossible under our
assumption.
Suppose E has multiplicative reduction at ℘, i.e. x specializes to a cusp
of X̃1(N). Recall the notation of τi and xi, 1 ≤ i ≤ d, in Section 2. Then
τi(K) is also a cubic field with prime ideal τi(℘) over p and residue field
ki = k. And τi(E) also has multiplicative reduction at τi(℘). This means
all the images x1, . . . , xd of x specialize to cusps of X̃1(N). Let c1, . . . , cd
be the cusps such that
xi ⊗ Fp = ci ⊗ Fp, 1 ≤ i ≤ d
We know all the cusps of X1(N) are defined over Q(ζN ) [18]. Let ℘′ be
a prime in Q(ζN ) over p. We also know p ramifies in Q(ζN ) if and only
if p|N . So e℘′(Q(ζN )/Q) = 1 under our assumption p - N . Therefore by
Lemma 3.3, the specialization map
Ψ : J1(N)(Q(ζN ))tor −→ J1(N)(Fp)
is injective.
cyclic torsion of elliptic curves 669
Since GonC(X1(N)) > d, then by Lemma 2.1, the map
Φ : X1(N)(d)(Q(ζN )) −→ J1(N)(Q(ζN ))
is injective.
We know x1 + · · · + xd is Q-rational and J1(N)(Q) is finite, so [x1 +
· · ·+xd−d∞] is in J1(N)(Q(ζN ))tor. By Theorem 3.4, the difference of two
cusps of X1(N) has finite order in J1(N). So [c1 + · · ·+ cd − d∞] is also in
J1(N)(Q(ζN ))tor. Therefore Ψ◦Φ(x1+· · ·+xd) = Ψ◦Φ(c1+· · ·+cd) implies
x1 + · · ·+ xd = c1 + · · ·+ cd since Ψ ◦Φ is injective. This is a contradiction
because we assume x is a noncuspidal point.
Therefore E has good reduction at ℘. 
4. Proof of Theorem 1.2
If N = 49, 40, 25 or 22, as is seen in Table 3.1, J1(N)(Q) is finite. By
Proposition 2.2 and 2.3, we know GonC(X1(N)) > 3. Let K be a cubic field
and ℘ a prime ofK over 3. Suppose x = (E,±P ) ∈ Y1(N)(K). Therefore by
Lemma 3.5, E has good reduction at ℘. By Theorem 2.5, the reduction P̃
of P is a k-rational point of order N in the elliptic curve Ẽ over k = OK/℘.
In the case N = 49 or 40, Ẽ(k) cannot have a point of order N since
N > (1 +
√
33)2 ≈ 38.4. Now consider the case N = 25 or 22. If k = F3
or F32 , then Ẽ(k) cannot have a point of order N since N > (1 +
√
32)2.
If k = F33 , suppose Ẽ(k) has a point of order N , then Ẽ(k) ∼= Z/NZ since
Nm > (1 +
√
33)2 for any m > 1. But by Theorem 2.6, |Ẽ(k)| 6= N (t = 3
for N = 25, t = 6 for N = 22). Thus, in any of the four cases, we have a
contradiction. So Z/NZ is not a subgroup of E(K)tor.
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Jian Wang
College of Mathematics
Jilin Normal University
Siping, Jilin 136000, China
E-mail: blandye@gmail.com


On the cyclic torsion of elliptic curves over cubic number fields (II)

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